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arXiv · 2609.02384

The truncated octahedron minimizes surface area among parallelohedra of equal volume

Annalisa Cesaroni·Matteo Novaga
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Abstract

We prove that the regular truncated octahedron uniquely minimizes surface area among all parallelohedra of fixed volume. Equivalently, every three-dimensional parallelohedron $P$ satisfies \[ \frac{\mathcal H^2(\partial P)}{|P|^{2/3}} \ge \frac{3(1+2\sqrt3)}{4^{2/3}}, \] with equality if and only if $P$ is similar to the regular truncated octahedron. Among the non-truncated Fedorov types we prove a stronger sharp bound, attained uniquely by the regular rhombic dodecahedron.

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Annalisa Cesaroni, Matteo Novaga. 2026-09-02. The truncated octahedron minimizes surface area among parallelohedra of equal volume. https://arxiv.org/abs/2609.02384

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